Third post-Newtonian dynamics of compact binaries: Equations of motion in the center-of-mass frame

TL;DR

This paper derives the third post-Newtonian order equations of motion for compact binary systems in the center-of-mass frame, providing explicit expressions for their dynamics, conserved quantities, and stability analysis.

Contribution

It presents the 3PN-accurate equations of motion and Lagrangian in the center-of-mass frame, linking harmonic and ADM formalisms, and analyzes circular orbit stability.

Findings

Derived explicit 3PN center-of-mass equations of motion.

Established equivalence between harmonic and ADM formalisms.

Analyzed stability of circular binary orbits at 3PN order.

Abstract

The equations of motion of compact binary systems and their associated Lagrangian formulation have been derived in previous works at the third post-Newtonian (3PN) approximation of general relativity in harmonic coordinates. In the present work we investigate the binary's relative dynamics in the center-of-mass frame (center of mass located at the origin of the coordinates). We obtain the 3PN-accurate expressions of the center-of-mass positions and equations of the relative binary motion. We show that the equations derive from a Lagrangian (neglecting the radiation reaction), from which we deduce the conserved center-of-mass energy and angular momentum at the 3PN order. The harmonic-coordinates center-of-mass Lagrangian is equivalent, {\it via} a contact transformation of the particles' variables, to the center-of-mass Hamiltonian in ADM coordinates that is known from the post-Newtonian ADM-Hamiltonian formalism. As an application we investigate the dynamical stability of circular binary orbits at the 3PN order.